Dmitry Malinin

Elliptic curves and Abelian varieties: some links to group representations, geometry and number theory.

We study some geometric and arithmetic properties of Abelian varieties, relatedgroup representations, Hopf algebras and quantum groups. In particular, representations attached to torsion points on supersingular elliptic curves give an infinite number of solutions of the tame inverse Galois problem, and the properties of formal groups and the action of the inertia group on the Tate module of the associated representations, can be generalized for Abelian varieties and supersingular Abelian surfaces with some extra conditions. We study Galois cohomology of related arithmetic groups. There is a connection between the study of special classes of coherent sheaves on elliptic curves and their degenerations with solutions of the classical Yang-Baxter equation; in particular, this theory contributes to classification of indecomposable vector bundles on elliptic curves, certain matrix problems and Frobenius Lie algebras.